Right now this matchup is slightly in supergirl's favor. The zoning battle is in her favor bc she can hit you with trait and recover in time to dodge an omega beam from fullscreen. Footsie range is in her favor as well since darkseid doesn't have anything to challenge her b1 and his anti airs get stuffed pretty easily if the supergirl knows how to space her j1 correctly. You have to call the parademon when she runs out of trait, that's key on this matchup bc this is the only time you'll be able to safely call a parademon in neutral, and you need to captalize on those opportunities. To make her run out of it you'll have to play the zoning game a bit, even tho this is a losing scenario, it's for a bigger purpose.
Careful with trait setups, most people respect it but when you fight a supergirl that understands the matchup they'll neutral jump float cancel, or air dash away and punish you for that, you can blow them up for that too, but you'll have to make the right read. Try to avoid air teleports bc she can hit you out of it for huge punishes with j2 teleport.
While darkseid does outdamage supergirl, they both have the same problem: they can't land huge punishes on moves that are below -14 on block. So everytime supergirl blocks a stomp she can deliver big dmg, but when darkseid goes for the overhead she can't. Same applies for ds, when supergirl tries to go for the overhead grab he can punish with big dmg, but while he can punish most of her neutral options, a d1 palm is usually all he can get. Projectile trades are in his favor but they won't be trading very often. Overall I feel like the damage department is pretty even in this matchup tbh.
Most supergirls don't know how to play this matchup, they punish stuff with b12 breath, air teleports with d2s, waste trait in combos, sit still watching you call a parademon and mix them up, try to throw random teleports to catch you off guard, try to pressure you with unsafe normals... that doesn't work. But when she is played solid and optimally she can effectively control the pace of the matchup. With the patch coming I expect this to be a 6-4 in her favor. Rn I'd say it's a 5,5-4,5 for her.